As Steve says, due to charging considerations, use the same type of battery you use for the house battery bank, In other words, if the house batteries are wet, gel or AGM use wet, gel or AGM, respectively for the starting battery. I have used both dedicated starting batteries and deep cycle bateries. For your small engine it won't really matter which you use if you use something in the grp 24 size range.

Now the wiring. I like to connect the positive output of the alternator directly to the battery positive terminal and the battery negative to a ground point on the engine. An alternative is to connect the alternator output the the battery switch and from there to the start battery. That connection allows you to completely de-energize the engine for working on it. Do install a battery switch (preferably a 1-2-both switch) between the battery and the starting circuits. Connect the starting battery to the "1" pole and the house bank to the "2" pole. Normal starting is done with the switch set at "1", but if the start battery is low, you can switch to the house bank or all the batteries for starting. Finally, I like battery combiners for connecting the house start and house batteries for starting purposes. With the alternator wired directly to the starting battery you don't have to worry about damaging the alternator by changing battery switches when the engine is running. You can change the switches to any setting you want. The combiner treats your house and starting batteries as one bank for charging purposes, but separates the batteries during discharge so you won't inadvertently discharge the start battery. I strongly suggest that you use actual battery switches which are make-before-break, so you can change switch positions without shutting your engine down accidentally. I like to install a 1-2-both switch for the engine circuit and a separate on-off switch for the house ban